Split 345 into it's prime factors and if one appears twice you can take that factor out of the square root. To find the prime factors try dividing by ascending prime numbers:

345 =3x 115 (so 3 is a prime factor)

115 =5x 23 (so 5 is a prime factor)

23 =23x 1 (so 23 is a prime factor)

As no prime factor occurs more than once cannot be simplified. Your answer is fully simplified

If, for example you had that could be simplified because 32 has prime factors of 2^5 which means you can take out two lots of two (ie: 4) and leave the sqrt2 behind:

In general it's because of the rules saying and